Discrete-time Waveform Relaxation Methods Of Nonlinear Differential-Algebraic Equations

Many problems in areas of science and engineering can be modeled by differential algebraic systems. Methods commonly used in the solution of differential-algebraic equation are an extension of numerical methods for solving ordinary differential equations. In 1980's, E.Lelarasmee et.al introduced waveform relaxation approach to solve the dynamical systems which are described by a system of mixed implicit algebraic-differential equations. Theoretical and computational studies show the method to be efficient and reliable. For the numerical solution of differential-algebraic equation waveform relaxation method provides a new way. After that, Parallel and decoupling properties of waveform relaxation approach begin to get more attentions.Waveform relaxation (WR) is a decoupling technique of large systems, which is first applied in circuit simulation. It is sometimes called dynamic iteration. The iterative equation for the discrete time is called as discrete-time dynamic iterative process. The key idea of waveform relaxation algorithm is to decompose original large systems into several subsystems, and then solve for each subsystem.In this paper we first review the background to the selection of waveform relaxation algorithm and introduce the history and development of this approach research. We mainly study the convergence conditions of the discrete-time waveform relaxation algorithm. In section two, we first discuss the discrete-time waveform relaxation method for an implicit system of nonlinear differential-algebraic equations which is derived by the backward-differentiation formulas (BDFs) and give some conclusions for the convergence of the methods. Then in section three, by discretising continuous-time waveforms using one-leg methods and linear multistep methods, we study the discrete-time waveform relaxation solutions, and some convergence conclusions are achieved. The proof is based on the properties of matrices. Under some assumptions, we get the convergence interval of the time step. By use of the one legθ-method and the linearθ-method, we obtain some conclusions for the convergence of the discrete waveforms of nonlinear integral-differential-algebraic equations in section four. Numerical experiments are provided to illustrate the theoretical results in section five.